Isoperimetric problem using the Gagliardo-Nirenberg-Sobolev

Question

Based in this post G-N-S $\Rightarrow$ Isoperimetric Inequality in Euclidean Space. With the Isoperimetric Inequality in $\mathbb{R}^{n}$. How can answer the following question: Between all open bounded subsets of $\mathbb{R}^{n}$ with fixed volume ,which one has the smallest surface area? I know which the answer is the sphere, but i don't know how prove it. Any tips?

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With the tips of Giuseppe and Guy , i imagine which the answer of my answer is prove that the inequality in the tagged post with the constant

$C=\frac{\mathcal{L}^{n-1}(\mathbb{S}^{n})}{\mathcal{L}^{n}(\mathbb{B}^{n})^{\frac{n-1}{n}}}$

right?

Answer 1

From this book by Cedric Villani Old and New page 571 you have that $$ \frac{|\partial B^n|}{|B^n|^{\frac{n-1}{n}}}\le \frac{|\partial A|}{|A|^{\frac{n-1}{n}}} $$

Now what happens if $|A|= |B^n(r)|?$ clearly it follows that $$\color{red}{|\partial B^n(r)|\le |\partial A| ~~~\text{which prove your statement}}$$